MA conference opening lecture

What is the source?

• Dr David Bedford, The Tuesday Boy Problem, Mathematical Association, Keele University, 10 April 2012.

What's the big idea?

• Conditional probability and variations on the Monty Hall Problem – investigating the counter-intuitive nature of probability and how information changes the probability of events.

What is their background?

• Dr David Bedford, School of Computing and Mathematics, Keele University.

What are they saying?

Question: 'I have two children, One is a boy born on a Tuesday. What is the probability I have two boys?'
(Attributed to Gary Foshee ‘G4G9’2010)
Is it:
a) a third?
b) a half?
c) don’t know?
d) don’t care?

Discussions about this have raged at: www.AlexBellos.com and you can see an outline at http://wmbriggs.com/blog/?p=2553

Let's rewind and look at a simpler problem: "Mr Smith has two children. At least one of them is a boy. What is the probability that both children are boys?"
Is it:
a) a third?
b) a half?
c) don’t know?
d) don’t care?
We can make progress by creating a table showing the four, (presumed equally likely) possibilities. I've greyed out the girl, girl possibility which is counter to the data in the question.

Younger Child	Older Child
Girl	Girl
Girl	Boy
Boy	Girl
Boy	Boy

Since we know that it can’t be the combination Girl, Girl we can conclude the probability of two boys must be one third.

Let’s subtly re-cast the problem: "Mr Smith has two children. The older child is a boy. What is the probability that both children are boys?"

Younger Child	Older Child
Girl	Girl
Girl	Boy
Boy	Girl
Boy	Boy

Hmm. Now the probability seems to be a half.

Now return to the '… On a Tuesday' problem.
To make progress create a grid showing all the possible days of the week for first born,and second born. Count the possible scenarios and see how many have two boys.h

Homer Simpson is a father of two. We meet him walking along the street with a boy whom he proudly introduces as his son. What is the probability that Mr Simpson has two boys?

Is it:
a) a third?
b) a half?
c) don’t know?
d) don’t care?

When we get information about one specific child, then the probability changes.

Mr Smith has two children. At least one is a boy with property P where this property occurs independently among boys with probability p. What is the probability that Mr Smith has 2 boys?

Younger Child Older Child probability
Girl Girl 1/4
Girl Boy with P p/4
Girl Boy without P

Boy Girl
Boy Boy

With some simplification we get P(two boys) = (2 – p) / (4 – p)

Watch the Kevin Spacey video below.



Notice it’s important that we assume in the classical Monty Hall problem that he does the same thing every time.

Things get interesting if you vary the number of doors, choices and reveals.

The two-stage Monty Hall allows the contestant to choose one from four and then reveal, switch or stick and then reveal a goat and play again. This is well explained using eight small counters.
The current game show 'Deal or No Deal' offers interesting ground, not because it is a Monty Hall problem (the boxes are opened without knowledge of their contents, so there is no additional information transmitted) but in the offers that the banker makes. It's particularly interesting to reverse the values of the boxes to be costs rather than prizes and the banker's offer to be the banker's charge: then you get insurance!

Where can I learn more?

http://www.m-a.org.uk/jsp/index.jsp?lnk=100
www.mathsjam.com

What next?

Link to GCSE curriculum: there's only a bit of conditional probability required for GCSE, but the use of tables of outcomes and tree diagrams in an engaging context may well help you make a memorable lesson.
Link to S1 chapter 5: probability.

*How will this change your practice?
*What do you or we need to do differently to make use of this idea?
*How will it improve learning?

Pay particular attention to the last question: if you can't see how it improves learning, say so.